Petr Vrána

Hamilton cycles in 5-connected line graphs

A conjecture of Carsten Thomassen states that every 4-connected line graph is hamiltonian. It is known that the conjecture is true for 7-connected line graphs. We improve this by showing that any 5-connected line graph of minimum degree at least 6 is hamiltonian. The result extends to claw-free graphs and to Hamilton-connectedness.

How Many Conjectures Can You Stand? A Survey

We survey results and open problems in hamiltonian graph theory centered around two conjectures of the 1980s that are still open: every 4-connected claw-free graph (line graph) is hamiltonian. These conjectures have lead to a wealth of interesting concepts, techniques, results and equivalent conjectures.

On stability of Hamilton-connectedness under the 2-closure in claw-free graphs

We show that, in a claw-free graph, Hamilton-connectedness is preserved under the operation of local completion performed at a vertex with 2-connected neighborhood. This result proves a conjecture by Bollobas et al.

Contractible subgraphs, Thomassen's conjecture and the dominating cycle conjecture for snarks

We show that the conjectures by Matthews and Sumner (every 4-connected claw-free graph is Hamiltonian), by Thomassen (every 4-connected line graph is Hamiltonian) and by Fleischner (every cyclically 4-edge-connected cubic graph has either a 3-edge-coloring or a dominating cycle), which are known to be equivalent, are equivalent to the statement that every snark (i.e. a cyclically 4-edge-connected cubic graph of girth at least five that is not 3-edge-colorable) has a dominating cycle. We use a refinement of the contractibility technique which was introduced by Ryjacek and Schelp in 2003 as a common generalization and strengthening of the reduction techniques by Catlin and Veldman and of the closure concept introduced by Ryjacek in 1997.

Thomassen's conjecture implies polynomiality of 1-Hamilton-connectedness in line graphs

A graph G is 1-Hamilton-connected if G−x is Hamilton-connected for every x∈V(G), and G is 2-edge-Hamilton-connected if the graph G+ X has a hamiltonian cycle containing all edges of X for any X⊂E+(G) = {xy| x, y∈V(G)} with 1≤|X|≤2. We prove that Thomassens conjecture (every 4-connected line graph is hamiltonian, or, equivalently, every snark has a dominating cycle) is equivalent to the statements that every 4-connected line graph is 1-Hamilton-connected and/or 2-edge-Hamilton-connected. As a corollary, we obtain that Thomassens conjecture implies polynomiality of both 1-Hamilton-connectedness and 2-edge-Hamilton-connectedness in line graphs. Consequently, proving that 1-Hamilton-connectedness is NP-complete in line graphs would disprove Thomassens conjecture, unless P = NP.

Closure, clique covering and degree conditions for Hamilton-connectedness in claw-free graphs

Abstract We strengthen the closure concept for Hamilton-connectedness in claw-free graphs, introduced by the second and fourth authors, such that the strong closure G M of a claw-free graph G is the line graph of a multigraph containing at most two triangles or at most one double edge. Using the concept of strong closure, we prove that a 3-connected claw-free graph G is Hamilton-connected if G satisfies one of the following: (i) G can be covered by at most 5 cliques, (ii) δ ( G ) ≥ 4 and G can be covered by at most 6 cliques, (iii) δ ( G ) ≥ 6 and G can be covered by at most 7 cliques. Finally, by reconsidering the relation between degree conditions and clique coverings in the case of the strong closure G M , we prove that every 3-connected claw-free graph G of minimum degree δ ( G ) ≥ 24 and minimum degree sum σ 8 ( G ) ≥ n + 50 (or, as a corollary, of order n ≥ 142 and minimum degree δ ( G ) ≥ n + 50 8 ) is Hamilton-connected. We also show that our results are asymptotically sharp.

Rainbow connection and forbidden subgraphs

A connected edge-colored graph G is rainbow-connected if any two distinct vertices of G are connected by a path whose edges have pairwise distinct colors; the rainbow connection number rc ( G ) of G is the minimum number of colors such that G is rainbow-connected. We consider families F of connected graphs for which there is a constant k F such that, for every connected F -free graph G , rc ( G ) ? diam ( G ) + k F , where diam ( G ) is the diameter of G . In this paper, we give a complete answer for | F | ? { 1 , 2 } .

A Closure for 1-Hamilton-Connectedness in Claw-Free Graphs

A graph G is 1-Hamilton-connected if G-x is Hamilton-connected for every vertex x∈VG. In the article, we introduce a closure concept for 1-Hamilton-connectedness in claw-free graphs. If Gi¾? is a new closure of a claw-free graph G, then Gi¾? is 1-Hamilton-connected if and only if G is 1-Hamilton-connected, Gi¾? is the line graph of a multigraph, and for some x∈VG, Gi¾?-x is the line graph of a multigraph with at most two triangles or at most one double edge. As applications, we prove that Thomassens Conjecture every 4-connected line graph is hamiltonian is equivalent to the statement that every 4-connected claw-free graph is 1-Hamilton-connected, and we present results showing that every 5-connected claw-free graph with minimum degree at least 6 is 1-Hamilton-connected and that every 4-connected claw-free and hourglass-free graph is 1-Hamilton-connected.

On Forbidden Pairs Implying Hamilton‐Connectedness

Let X, Y be connected graphs. A graph G is -free if G contains a copy of neither X nor Y as an induced subgraph. Pairs of connected graphs such that every 3-connected -free graph is Hamilton connected have been investigated most recently in (Guantao Chen and Ronald J. Gould, Bull. Inst. Combin. Appl., 29 (2000), 25–32.) [8] and (H. Broersma, R. J. Faudree, A. Huck, H. Trommel, and H. J. Veldman, J. Graph Theory, 40(2) (2002), 104–119.) [5]. This paper improves those results. Specifically, it is shown that every 3-connected -free graph is Hamilton connected for and or N1, 2, 2 and the proof of this result uses a new closure technique developed by the third and fourth authors. A discussion of restrictions on the nature of the graph Y is also included.

On cycle lengths in claw-free graphs with complete closure

We show a construction that gives an infinite family of claw-free graphs of connectivity @k=2,3,4,5 with complete closure and without a cycle of a given fixed length. This construction disproves a conjecture by the first author, A. Saito and R.H. Schelp.